You should get a small server for your classroom if the school doesn't have a server you can use.
I would recommend using synology. They have decently priced systems that are incredibly easy to setup and use.
Something like this would be good enough.https://www.amazon.com/Synology-5bay-Disk-Station-Diskless/dp/B071K9HS6G...
Just export as a ProRes422(HQ) or a DNxHD 175x.
I always master as PRHQ unless the source material's bit rate is greater than a PRHQ. For that, I usually default to using a PRXQ or a DPX sequence.
It really depends on the work you're doing. Avid is the standard in most professional post houses where the work never leaves the confines of the office.
Trying to do freelance work with Avid can be such a huge pain though. Especially if you're not the one doing the online work and your client is in another city.
I would say that if you're using an NLE just to ingest and review footage, I would use either DaVinci or Premiere. In my opinion those would work best for your situation.
*Side note* If you haven't updated your premiere this month, you should give it a shot. It might fix some of your issues.
Jesus. Can you just shut up.
Every time you say something it just suuuuuuuuuuuuuuucks.
There's not too many great blu ray authoring programs out there for super cheap.
You can get toast but you have to buy an upgrade to burn Blu Rays.
What I use at work and at home for screeners and such is Adobe Encore. It's part of CS6 but it still works pretty well and it's an easy workspace to navigate.
If you don't have creative cloud, you can most likely find it on the interwebs for free.
I feel bad for Mark Frost. He's never going to get credit for Twin Peaks.